XL International Symposium on Multiparticle Dynamics
Inclusive cross sections of
proton-proton and
proton-antiproton scattering
Victor A. Abramovsky
Natalia V. Radchenko
Novgorod State University
Russia
Inclusive cross sections and multiplicity
distributions are different for pp and pbarp
Pomeranchuk theorem:
total cross sections
differential elastic cross sections
elastic cross sections
the same
for pp and pbarp
It is also generally accepted that inclusive
cross sections and multiplicity distributions are the
same for pp and pbarp.
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Low Constituents Number Model (LCNM)
1. On the first step before the collision there is small number
of constituents in initial hadrons. In every hadron this is
component either with only valence quarks or with valence
quarks and one additional gluon.
2. On the second step the hadrons interaction is carried out by
gluon exchange between the valence quarks and initial
gluons. The hadrons gain the color charge.
3. On the third step after interaction the colored hadrons move
apart and when the distance between them becomes larger
than the confinement radius, the lines of color electric field
gather into the string. This string breaks out into secondary
hadrons.
(Abramovsky, Kancheli 1980, Abramovsky, Radchenko 2009)
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Three types of inelastic subprocesses
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Results in LCNM
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Pseudorapidity density at LHC energies
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The importance of inclusive cross sections
• From the LCNM it follows, that multiplicity distributions are
different for pp and pbarp because of subprocess with three quark
strings in pbarp (blue line).
• Three quark strings produce events with high multiplicity in tail of
distribution.
• In order to make the difference more visible, we have to use
observable n  Pn instead of Pn
• n  Pn is measured independently of Pn in inclusive measurements
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Inclusive cross sections in events with fixed
number of particles
Topological inclusive cross section of one charged particle production

2
d 3 nincl
1
1
3
2  2E 3 
d n1m A2nm


n  1! m0 m!
d p
Total inclusive cross section and its normalization
d 3 incl
d 3 nincl
3
3
2  2E 3   2  2E 3
d p
d p
n 0

3
d
 p
d 3 incl
d3p
 n  nsd
Normalization of topological inclusive cross section
1  1
 n    d n  m A2n  m
n ! m 0 m !
3 incl
d
n
3
d
p
 nn

3
d p
1
 nsd
2
3 incl
d
n
n
3
d
p

n
 n Pn

3
nsd
d p

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Inclusive cross sections in bins
UA5 Coll. gave data in 9 bins of charged multiplicities:
2 ≤ n ≤ 10, 12 ≤ n ≤ 20, … 72 ≤ n ≤ 80 and n ≥ 82.
We define inclusive cross section in this bins
10
d 3 nincl
d 3 (1) incl
 3 ,
3
d p
n2 d p
20
d 3 nincl
d 3 ( 2) incl
  3 , ...
3
d p
n 12 d p
9
d 3 (i ) incl
i 1
d3p


d 3 nincl
d 3 (9) incl
  3 ,
3
d p
n 82 d p
d 3 incl

d3p
Inclusive cross sections in bins are normalized as follows
3 ( i ) incl
d

3
nsd
( i ) nsd
d
p


n
P

n


n

3
d p
n in bin
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Difference in inclusive cross sections of pp and pbarp
3
d
 p
d p
3
d 3 (ppi ) incl
3
d p
d 3 (pip) incl
3
d p
  d d 2 p
  d d p
2
d 3 (ppi ) incl
d d p
2
d 3 (pip) incl
d d p
2
  d
  d
d (ppi ) incl
d
d (pip) incl
d
( i ) nsd
 n pp

(1)
 n p(ip) nsd
(2)
d (i ) incl
d 3 (i ) incl
2
  d p
d
d d 2 p
Ratio of (1) to (2) gives
 d
d (ppi ) incl
d

(i )
n pp
n p(ip)
 d
d (pip) incl
d
(3)
Solution of the integral equation (3) (perhaps, the only solution)
d (ppi ) incl
d

(i )
n pp
d (pip) incl
n p(ip)
d
(4)
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Inclusive cross sections in different bins
(i )
n p(ip) n pp
2 ≤ n ≤ 10
0.76 ± 0.01
12 ≤ n ≤ 20
0.86 ± 0.01
22 ≤ n ≤ 30
0.99 ± 0.01
32 ≤ n ≤ 40
1.09 ± 0.01
42 ≤ n ≤ 50
1.10 ± 0.01
52 ≤ n ≤ 60
1.18 ± 0.01
62 ≤ n ≤ 70
1.35 ± 0.02
72 ≤ n ≤ 80
1.45 ± 0.02
n ≥ 82
1.26 ± 0.02
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Inclusive cross sections summed over all bins
We obtained the numerical
value of relation
d pincl
p
d incl
pp
d
d
by summing over all nine
bins of multiplicity.
For   2.5
d incl
pp
d incl
pp
d
d
 1.12  0.01
This result will be used on
the next slides.
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Inclusive cross section with transverse momentum
From the AGK cancellation rules it follows the factorization of
transverse dependence in inclusive cross section.
1 d 2 incl
d incl
 f  p 
2 p d dp
d
Returning to formulas (1) and (2) we can write down
d 3 (ppi )incl
d d p
2

n p(ip) d 3 (pip)incl
(i )
n pp
d d 2 p
From this relation it can strictly be proved that
so we can obtain
2 incl
1 d  pp
2 p d dp
2 incl
d incl
1 d  pp
pp

2 p d dp
d
f pp  p   f pp  p 
d incl
pp
d
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Experimental evidences of difference in pp and pbarp
ATLAS Coll.
ALICE Coll.
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Conclusion
• We think that ATLAS Coll. discovered a new
effect – the difference in multiparticle production
in pp and pbarp interactions. ALICE Coll.
confirmed this effect.
• We propose to experimentalists to measure
inclusive cross section in bins with high
multiplicities, where the difference will be the
most evident.
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Acknowledgements
• V.A.A. gratefully acknowledges financial support
from the RFBR, grant 10-02-68536-z.
• N.V.R. gratefully acknowledges financial support
from the Ministry of education and science of the
Russian Federation, grant P1200.
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